Limits Continuity And Differentiability

Left and right limits of $f$ at $x=a$ are equal and they are not equal to $f(a)$

Both left and right limits of $f$ at $x=a$ exist and are not equal

The function $f(x)$ is continuous at $x=a$

The function $f(x)$ has a simple discontinuity at a point other than $a$